Literal listener (tries to infer the probability \(\phi\) of getting a blue gumball given an utterance \(u\)):
\[P_L(\phi \mid u; \theta) \propto P(\phi) \left( {0.95} \times \mathbb{1}[\phi > \theta_{u}] + 0.05 * P_{uniform}(\phi; 0,1) \right)\] (The term \(0.05 * P_{uniform}(\phi; 0,1)\)) corresponds to the noise term which assigns a small non-zero probability to all \(\phi\) idependent of the actual utterance.) We assume that the prior \(P(\phi)\) is uniform.
Pragmatic speaker (marginalizes over all possible values of \(\theta\)):
\[P_S(u \mid \phi, condition) \propto \int P(\theta) \exp\left(\lambda * \left(\log P_L(\phi \mid u; \theta) - c(u, condition)\right)\right) d\theta \]
Costs:
\[ c(u, condition) = \begin{cases} 0 &\quad\text{if } u \text{ is one of the utterances in } condition\\ c_{u} &\quad\text{otherwise} \\ \end{cases} \]
Prior over thresholds \(\theta\):
\[P(\theta_u) = Beta(\alpha_u, \beta_u)\]
Estimated parameters:
\(\alpha_u, \beta_u \sim Uniform(0,30)\), \(c_u\sim Uniform(0,5)\), \(\lambda \sim Uniform(.1,3)\)
Left column: Experimental data.
Middle column: Model predictions with parameters estimated from all conditions.
Right column: Model predictions with parameters estimated from all conditions but the current one.
## R^2: 0.962100734845317
## R^2: 0.945208331407077
## R^2: 0.893958743002193
## R^2: 0.881150419543416
## R^2: 0.92887618625227
## R^2: 0.875702859306288
## R^2: 0.911223847778544
## R^2: 0.891817769134906
## R^2: 0.92681575326815
## R^2: 0.926917442031885
## R^2: 0.884452278220882
## R^2: 0.87354581218351
## R^2: 0.803185003839032
## R^2: 0.72348001033727
## R^2: 0.885954531882677
## R^2: 0.886021512162489
## R^2: 0.916130676480786
## R^2: 0.934234780070048
## R^2: 0.89054350714818
## R^2: 0.889419195614308
## R^2: 0.831921319476047
## R^2: 0.858918304337029
## R^2: 0.891647987894687
## R^2: 0.849749570608412
## R^2: 0.853554107925385
## R^2: 0.885507294384094
## R^2: 0.822272345719847
## R^2: 0.790476182241034
## R^2: 0.891972040221257
## R^2: 0.893312880931263
## R^2: 0.848189269538628
## R^2: 0.838660578889502